How things flow through committees is up to House leadership, which these days means the Republican Party. If the minority party would like to rescue legislation from that gulag, there is a tool at their disposal. It's called a discharge petition.
A discharge petition is what it sounds like: a petition that must be signed by a majority of House members that discharges the legislation from committee. As you can imagine, that doesn't happen very often. Since 1999, only two items have been successfully discharged from the committee gulag out of 107 attempts. Most fall far short of the required 218 votes. On average, discharge petitions get 143 signatures.
On Friday, though, the House got its 218th signature on a petition for the first time since January 2002. In that case, it was a bill focused on campaign finance reform, the Bipartisan Campaign Reform Act of 2002, that ended up becoming law. In this case, it's legislation that would reauthorize the Export-Import Bank. (The what?)
One reason that discharge petitions are rarely approved is that they generally require bipartisanship. After all, the majority party can generally move things through the process as it sees fit. It's the minority party that needs to force the issue. If you look at the discharge petitions since the Republicans retook the House after the 2010 elections, you see that nearly all of the signers were Democrats.
One obvious exception is that second one in the top row. That one was a petition backed by a number of the Republican members of Congress who have been giving the Republican establishment such a hard time since that election cycle. The bill to be discharged would have ensured military pay in the event of a funding gap (i.e. government shutdown). A sure sign it was backed by the far-right caucus? Reps. Louie Gohmert and Steve King were two of the first three signers.)
That contingent willing to challenge the Republican leadership has grown in power to the extent that John Boehner is resigning as speaker of the House and his successor isn't clear. It was strong enough to block the Ex-Im bank's reauthorization in the first place — so you won't see Gohmert's name on the new petition. Instead, it's more mainstream members of the Republican caucus, joining with Democrats, to force the issue.
As we said, this is rare. Meaning the discharge petition. And the bipartisanship. And the sort-of-victory for the Republican establishment.